I have several of those moments for each game really. Way too many to list here and too hard to pick one - but here are a few :
GTA SA - Taking road trips from city to city, cruising along the highway as the sun was setting in the background while 808 State - Pacific State was playing on SF-UR.
GTA VC - Eating a portion of fish and chips while flying a sparrow across downtown while listening to Wildstyle
I know VCMP shouldn't count but one of the best moments ever was when I was cruising down the Downtown main road in VCMP with tons of players, bullets whizzing everywhere and exploding helicopters and cars scattered everywhere all while listening to Wildstyle.
GTA III - Walking out of the Sex club 7 alley like a badass while the mission complete tune was playing. Being so f*cking excited after unlocking Staunton Island.
GTA CTW - Cruising down the FIA Highway in a Sabre GT while listening to Deadmau5.
GTA IV - Gasping in amazement at how even the sh*ttiest of vehicles looked beautiful due to the graphics. Cruising down Algonquin in the rain with JnR. The music coupled with the rain made me feel so cosy.
Sorry for the long post but these are only a few. I have tons of memorable moments that I could cry just thinking of for each and every GTA game I've played. This post has been edited by Rerxelend on Sunday, Feb 19 2012, 19:39